﻿ Analytic geometry Planes in 3d Formulas
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Plane forms

Point direction form: $$a(x-x_1) + b(y-y_1) + c(z-z_1) = 0$$ where P(x1,y1,z1) lies in the plane, and the direction (a,b,c) is normal to the plane. General form: $$Ax + By + Cz + D = 0$$ where direction (A,B,C) is normal to the plane. Intercept form: $$\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$$ this plane passes through the points (a,0,0),(0,b,0) and (0,0,c). Three point form: $$\begin{vmatrix} x-x_3 & y-y_3 & z-z_3 \\ x_1-x_3 & y_1-y_3 & z_1-z_3 \\ x_2-x_3 & y_2-y_3 & z_2-z_3 \end{vmatrix} = 0$$ Normal form: $$x\,\cos \alpha + y\,\cos\beta + z\,\cos\gamma = p$$ Parametric form: \begin{aligned} x &= x_1 + a_1\,s + a_2\,t \\ y &= y_1 + b_1\,s + b_2\,t \\ z &= z_1 + c_1\,s + c_2\,t \end{aligned} where the directions (a1,b1,c1) and (a2,b2,c2) are parallel to the plane.

Angle between two planes:

The angle between planes A1x+B1y+C1z+D1=0 and A2x+B2y+C2z+D2=0 is: $$\alpha = \arccos \frac{A_1A_2 + B_1B_2 + C_1C_2} {\sqrt{A_1^2 + B_1^2 + C_1^2} \cdot \sqrt{A_2^2 + B_2^2 + C_2^2}}$$ The planes are parallel if and only if $$\frac{A_1}{A_2} = \frac{B_1}{B_2} = \frac{C_1}{C_2}$$

Equation of a plane

The equation of a plane through P1(x1,y1,z1) and parallel to directions (a1,b1,c1) and (a2,b2,c2) has an equation: $$\begin{vmatrix} x-x_1 & y - y_1 & z - z_1 \\ a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \end{vmatrix} = 0$$ The equation of a plane through P1(x1,y1,z1) andP1(x2,y2,z2)), and parallel to direction (a,b,c), has equation $$\begin{vmatrix} x-x_1 & y - y_1 & z - z_1 \\ x_2-x_1 & y_2-y_1 & z_2-z_1 \\ a & b & c \end{vmatrix} = 0$$ The equation of a plane through P1(x1,y1,z1) , P2(x2,y2,z2) and P3(x3,y3,z3) , has equation $$\begin{vmatrix} x-x_1 & y - y_1 & z - z_1 \\ x_2-x_1 & y_2-y_1 & z_2-z_1 \\ x_3-x_1 & y_3-y_1 & z_3-z_1 \end{vmatrix} = 0$$

Distance from point to plane

The distance of P1(x1,y1,z1) from the plane Ax+By+Cz+D=0 is $$d = \frac{Ax_1 + By_1 + Cz_1}{\sqrt{A^2 + B^2 + C^2}}$$

Intersection of two planes

The intersection of planes A1x+B1y+C1z+D1=0 and A2x+B2y+C2z+D2=0 is the line: $$\frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c}$$ where \begin{aligned} a &= \begin{vmatrix} B_1 & C_1 \\ B_2 & C_2 \end{vmatrix}~~ b = \begin{vmatrix} C_1 & A_1 \\ C_2 & A_2 \end{vmatrix}~~ c = \begin{vmatrix} A_1 & B_1 \\ A_2 & B_2 \end{vmatrix} \\ x_1&= \frac{b\begin{vmatrix}D_1& C_1 \\ D_2 & C_2 \end{vmatrix} - c\begin{vmatrix}D_1& B_1 \\ D_2 & B_2 \end{vmatrix} }{a^2 + b^2 + c^2} \\ y_1&= \frac{c\begin{vmatrix}D_1& A_1 \\ D_2 & A_2 \end{vmatrix} - a\begin{vmatrix}D_1& C_1 \\ D_2 & C_2 \end{vmatrix} }{a^2 + b^2 + c^2} \\ z_1&= \frac{a\begin{vmatrix}D_1& B_1 \\ D_2 & B_2 \end{vmatrix} - b\begin{vmatrix}D_1& A_1 \\ D_2 & A_2 \end{vmatrix} }{a^2 + b^2 + c^2} \end{aligned} If a=b=c=0, then the planes are parallel.
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